We have traveled to Orlando 5 times with our children since the youngest was 11 months old, we have always taken a full size car seat and stroller for each of them.
I would never rent a car seat from the hire car company - my two main reasons for this are:-
1.Imagine the scenario, you've just arrived in Florida, you are tired, kids are tired, you are now going to fit an unfamiliar car seat into an unfamiliar vehicle, how do you know it's fitted correctly? I'm not sure I would trust the people working in the parking garage to know that it's done right, your own seat is familiar to you, you'll fit it properly and know that your child is safe.
2. The car seats that I have seen in the parking garage for hire are the most grubby, stinky looking things you have ever seen, apart from them (possibly) being dirty you won't know the history of the seat, it may have been damaged and is still being rented?
We have always used the big plastic bags that the airlines provide and then they ask you to go to the oversized baggage area in the airport with them, this way they don't go through the airport on the conveyor belts so they don't get bashed that way.
We have never been charged any extra for them either.
As Elise79 said you need to make sure that you get your stroller checked at check in but don't hand it over until you are ready to board. Sometimes they come off and are waiting in the tunnel where you get off the plane - other times they are in the baggage hall, I would always ask the crew on the flight what they usually do as every airline is different.
HTH
Gems